Intuitively, causation seems to require not just a correlation, but a counterfactual dependence.

Intuitively this seems to mean that the number of inputs and the number of outputs can always be equal in each subswitch, but intuition does not prove this can be done nor does it tell us how to do so.

Intuitively at least, it seems that Newton's theory of motion entails a good many more truths than does, say, Aristotle's theory-despite the fact that both are known to have flaws.

Intuitively, NP is the set of all decision problems for which the instances where the answer is " yes " have efficiently verifiable proofs of the fact that the answer is indeed " yes ".

Intuitively then, the oracle machine can perform all of the usual operations of a Turing machine, and can also query the oracle for an answer to a specific question of the form " is x in A?

Intuitively, the distinction says merely that there is no canonical choice of where the origin should go in the space, because it can be translated anywhere.

Intuitively, it should be

Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e. g. the value assigned to " 1 or 2 " in a throw of a die should be the sum of the values assigned to " 1 " and " 2 ".

Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.

Intuitively, one should think of a + b as the " union " or the " least upper bound " of a and b and of ab as some multiplication which is monotonic, in the sense that a ≤ b implies ax ≤ bx.

Intuitively, one may think that the larger the weight for a criterion is, the more critical that criterion should be.

Intuitively, if a coastline looks smooth it should have dimension close to 1 ; and the more irregular the coastline looks the closer its dimension should be to 2.

Intuitively, if both coins are tossed the same number of times, the first coin should turn up fewer heads than the second one.

Intuitively, this corresponds to finding the amount of output each sector should produce given that we want 7 units of good A and 4 units of good B.

Intuitively, one might think that such a wave function should randomly ionize atoms throughout the cloud chamber, but this is not the case.

Intuitively, one would expect systems which collect more than one type of data to be more useful than systems which collect only one type of information ( such as single-purpose laboratory or 911 call-center based systems ), and be less prone to false alarms, and this appears to be the case.

Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (- π, π ) on the x-axis ; then bend the ends of this interval upwards ( in positive y-direction ) and move them towards each other, until you get a circle with one point ( the topmost one ) missing.

Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.

Intuitively, this means there are computational limits on what can be proven by computer programs.

Intuitively, theories of value must be important to ethics.

Intuitively, multiplying by a scalar r stretches a vector out by a factor of r. Geometrically, this can be visualized ( at least in the case when r is an integer ) as placing r copies of the vector in a line where the endpoint of one vector is the initial point of the next vector.

Intuitively this would seem to be folly because it would cool the iron.

Intuitively, the divergence ( denoted ∇•) of a vector field is a measure of flux diverging radially outwards from a point, so the negative is the amount piling up at a point, hence the rate of change of density in a region of space must be the amount of flux leaving or collecting in some region ( see main article for details ).

Schubert wrote, " Intuitively, iterated limiting identification might be regarded as higher-order inductive inference performed collectively by an ever-growing community of lower order inductive inference machines.

Intuitively we would expect it to be even more unlikely for all 5 marbles to be white.

Intuitively, represents a small perturbation in the index of T. By noting that and that is always between 0 and 1, can be used to ignore the floor function in the index.

Intuitively, one can think of the radical of I as obtained by taking all the possible roots of elements of I. Rad ( I ) turns out to be an ideal itself, containing I.

If, for example, we simply look at a curve in the real affine plane there might be singular P modulo the stalk, or alternatively as the sum of m ( m − 1 )/ 2, where m is the multiplicity, over all infinitely near singular points Q lying over the singular point P. Intuitively, a singular point with delta invariant δ concentrates δ ordinary double points at P. For an irreducible and reduced curve and a point P we can define δ algebraically as the length of where is the local ring at P and is its integral closure.

Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces.

Intuitively, a continuous curve in 2 or 3 ( or higher ) dimensions can be thought of as the path of a continuously moving point.

Intuitively, the continuous operator A never " lengthens " any vector more than by a factor of c. Thus the image of a bounded set under a continuous operator is also bounded.

Intuitively, the closer the point in question is to this center of mass, the more likely it is to belong to the set.

Intuitively, it is thought that there are more positive integers than perfect squares, since every perfect square is already positive, and many other positive integers exist besides.

Intuitively one wants to choose h as small as the data allow, however there is always a trade-off between the bias of the estimator and its variance ; more on the choice of bandwidth later.

) Intuitively, is a quine, a function that returns its own source code ( Gödel number ), except that rather than returning it directly, passes its Gödel number to and returns the result.

Intuitively, the reasoning is that, as web crawlers have a limit to how many pages they can crawl in a given time frame, ( 1 ) they will allocate too many new crawls to rapidly changing pages at the expense of less frequently updating pages, and ( 2 ) the freshness of rapidly changing pages lasts for shorter period than that of less frequently changing pages.

Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is no greater than a definite real number ; this bound is called the function's " Lipschitz constant " ( or " modulus of uniform continuity ").

Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows.

Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in D ; if the coefficient in D at z is negative, then we require that h has a zero of at least that multiplicity at z – if the coefficient in D is positive, h can have a pole of at most that order.

Intuitively, one can reason that Hick's Law has a logarithmic form because people subdivide the total collection of choices into categories, eliminating about half of the remaining choices at each step, rather than considering each and every choice one-by-one, requiring linear time.

Intuitively, if the leader was no better off than the follower, it would simply adopt a Cournot competition strategy.

Intuitively, this can be understood as follows: In quantum mechanics, the entropy of the joint system can be less than the sum of the entropy of its components because the components may be entangled.

Intuitively, an infinite graph has arbitrarily large finite subgraphs with any density less than its upper density, and does not have arbitrarily large finite subgraphs with density greater than its upper density.